This post is intended as a contribution to the (already admittedly very large) corpus of discussion surrounding the Sleeping Beauty paradox. The statement of the paradox (which I will simply refer to here as the Paradox) itself is assumed to be known, as it is widely available on the internet, including on this site.
When I first learned of the Paradox, I was an intuitive one halfer. This post is meant to describe the reasoning path that led me to becoming a one thirder.
Specifically, for reasons I will further detail below, I believe that much of the confusion on the resolution of the Paradox stems from a failure to differentiate between two clearly distinct questions:
Q1: The experiment has started, and Sleeping Beauty is woken up. What is the probability that the coin landed on heads?
Q2: The experiment has started, and Sleeping Beauty is woken up. As a rational individual, what value should she propose for the probability that the coin landed on heads?
I believe that the answer to Q1 is one half, and that the answer to Q2 is one third. Since the question posed in the Paradox is Q2, I am a one thirder.
Let me try to explain why.
Step 1: Preliminary Remarks
In certain situations, the fact that the experimental observer is capable of proposing a probability value for a certain event matters to the probability value she should rationally propose for that event. For instance, let us consider a situation where Sleeping Beauty is subjected to the following experiment:
Note, that similar to conditions in the Paradox, the full experimental process is transparently explained to Sleeping Beauty before she enters the experiment.
The experiment starts. A fair coin is tossed. Sleeping Beauty lies down on her mattress and explosive pillow. Sleeping Beauty is put to sleep. Some time elapses and Sleeping Beauty is woken up.
At this juncture, let us pause. There are two vastly different questions here:
Q3: What is the probability that the coin landed on heads?
Q4: What value should Sleeping Beauty, as a rational individual, propose for the probability that the coin landed on heads?
The answer to Q3 is one half. The answer to Q4 is zero.
I believe this is evident enough to not require a justification. Let us simply remark that the sentence “What value should Sleeping Beauty, as a rational individual, propose for the probability that the coin landed on heads?” has an implicit beginning which, made explicit, would read: “In the event that she is capable of proposing a value…”
Step 2: the Dispensable Airlock
Now let us return to an experimental set up somewhat closer to that of the Paradox.
The experiment starts. A fair coin is tossed. Sleeping Beauty lies down in the airlock. Sleeping Beauty is put to sleep. Some time elapses and Sleeping Beauty is woken up. She finds herself in the airlock. What value should she propose for the probability that the coin landed on heads
It depends on what the experimenters (which we assume here and throughout to be always truthful and transparent) told Sleeping Beauty about the experimental set up. They could have told her that either:
If H1 was the case, Sleeping Beauty should propose one third. If H2 was the case, Sleeping Beauty should propose one half.
Step 3: the Indispensable Airlock
Now let us modify that experimental set up to be as follows:
The experiment starts. A fair coin is tossed. Sleeping Beauty begins breathing though the astronaut-like apparatus. Sleeping Beauty lies down in the airlock. Sleeping Beauty is put to sleep. Some time elapses and Sleeping Beauty is woken up. This might or might not be the first time that Sleeping Beauty is woken up in the experiment - she doesn’t know.
Q5: What is the probability that the coin landed on heads?
Q6: What value should Sleeping Beauty, as a rational individual, propose for the probability that the coin landed on heads?
The answer to Q5 is one half. The answer to Q6 is one third.
Step 4: Looping Back
Why is Q6 equivalent to Q2, the question originally posed in the Paradox?
Compared to the experimental set-up of the Paradox, the Indispensable Airlock experimental set-up changes only one element: if the coin lands on heads, Sleeping Beauty will be killed, rather than left to sleep, on Tuesday. Because the death occurs so rapidly after the waking, nothing in the cognitive internal state of Sleeping Beauty has changed when she dies on Tuesday compared to what her cognitive internal state was when she fell back asleep on Monday.
Therefore, modifying the Paradox’ experimental set up to the Indispensable Airlock experimental set up is equivalent to stating that, if the coin lands on heads, Sleeping Beauty will be killed in her sleep, rather than left to sleep on Tuesday.
From the point of view of the probabilistic experiment, this is a distinction without a difference. Therefore Sleeping Beauty should answer one third to Q2, as she does to Q6.
I don't follow. In the airlock/death case, I think she should predict 0% heads. If she's wrong, she's about to die, but there's no benefit to predicting that. In the standard case, she should bet 1/3 or 1/2 depending on what experiences she cares about. 1/3 for cases where being wrong twice is more harmful than being wrong once, and 1/2 if the payouts are equal.
Specifying "as a rational individual" doesn't disambiguate these reasons for predicting.
I don't think so. Even in the heads case, it could still be Monday - and say the experimenter told her: "Regardless of the ultimate sequence of event, if you predict correctly when you are woken up, a million dollars will go to your children."
To me "as a rational individual" is simply a way of saying "as an individual who is seeking to maximize the accuracy of the probability value she proposes - whenever she is in a position to make such proposal (which implies, among others, that she must be alive to make the proposal)."
This is why it's important to specify what future impact the prediction has. "accuracy" has no meaning outside of resolution - what happens in the future.
Adding the result "$1M to your kids if you predict correctly" makes 1/2 the obvious and only choice. "Feel good about yourself if you survive" makes 0% the correct choice. Other outcomes can make 1/3 the right choice.
I do not agree that accuracy has no meaning outside of resolution. At least this is not the sense in which I was employing the word. By accurate I simply mean numerically correct within the context of conventional probability theory. Like if I ask the question "A dice is rolled - what is the probability that the result will be either three or four?" the accurate answer is 1/3. If I ask "A fair coin is tossed three times, what is the probability that it lands heads each time?" the accurate answer is 1/8 etc. This makes the accuracy of a probability value proposal wholly independent from pay-offs.
Cool, I think we've found our fundamental disagreement. I do not agree that "numerically correct within the context of conventional probability theory" is meaningful. That's guessing the teacher's password, rather than understanding how probability theory models reality.
In objective outside truth (if there is such a thing), all probabilities are 1 or 0 - a thing happens or it doesn't. Individual assessments of probability are subjective, and are about knowledge of the thing, not the thing itself. Probabilities used in this way are predictions of future evidence.
If you don't specify what evidence you're predicting, it's easy to get confused about what calculation you should use to calculate your probability.
All right - but here the evidence predicted would simply be "the coin landed on heads", no? I don't really the contradiction between what you're saying and conventional probability theory (more or less all which was developped with the specific idea of making predictions, winning games etc.) Yes I agree that saying "the coin landed on heads with probability 1/3" is a somewhat strange way of putting things (the coin either did or did not land on heads) but it's a shorthand for a conceptual framework that has firmly simple and sound foundations.